Very slow internet

[#1]

DO NOT CUT ANY WIRES OR TOUCH THE CONNECTORS. The current mess should be enclosed in a proper BT grey box, but these boxes are not sealed. The clear connectors you see are silicon filled crimps that protect the joints from the elements. The primary purpose of the outer box is to protect the inner cores of the cable from being exposed to direct sunlight and the driving elements.
I'll post more in a moment, wanted to be sure you saw this as soon as possible.

[#2]

Let me say at the outset that ANY attempt to mess with this or enclose it in anything other than the correct BT box is only likely to make it more obvious someone has been messing or damaged things they shouldn't.
Any person competent with a screwdriver should have been able to unscrew the original box and screw it back on the new soffit board. So your argument needs to be with the builder who should foot any bill from BT.
Your photo isn't quite clear enough on the detail but as those are two separate cables, the black sheathed incoming and the white one running down the wall, there is no way any of the wires can run straight through. Only one pair of wires is used for a single line, usually an orange and white pair which looks to be the case here. The green and black pair are generally only used for second lines.
I'd suggest your best bet is to get the builder to admit it's his responsibility (even if he subcontracted to a fascia company it's still his problem) and then to gently push the wires and connectors under the soffit a bit more without putting any strain on the wires or joints, so they are slightly better protected from the elements.
Having done that, if you can spot an Openreach van and engineer with a ladder nearby, is for your builder to offer him a few quid, and you copious tea and biscuits for the engineer to come and tidy it up for you. Hopefully he'll put a new box on and recrimp the joints. Alternatively get a competent person to scrounge or buy the correct box and crimps and do the job. Getting BT in will probably be the most expensive option.
With regards to your current fault, if you are syncing at 7000 with the exchange, the connections are OK at this instant, but those crimps can dry out with time. (Note  - only the Orange and White wires should be connected)

[#3]

Well several weeks on, and the nature of my problem has changed somewhat. I now get decent sync and throughput speeds on the whole (especially if I am connected into the master test socket: 5.5Mbps compared with around half that if I'm not in the master test socket, which I guess means that my internal house wiring is less than ideal). However, I get intermittent periods of downtime (typically lasting 10 - 20 minutes, where connectivity is either extemely slow or non-existent). The router doesn't think it has lost sync, and I can't run any speed tests. Then, all of a sudden, it starts working again.
I am thinking of replacing the faceplates on the two extension sockets in my house, just to rule them out of the equation as they seem quite old (one had a dead ant inside it). When I went to B&Q today, I see you can buy "master" phone sockets that look like the right thing, but I was not entirely sure if that would be the wrong thing to get, since you are only supposed to have one "master" socket in the house. My two "extensions" both come directly out the master socket (not plugged into the front, but wires coming out from the insides of the master socket). One goes only a few inches through a wall, while the other travels about 10 metres up and over two doors and round the side of a room. Can I buy the master faceplate from B&Q, or do can I get a "secondary" one?

[#4]

Has your master socket get a removable lower half and if so are the extensions connected to the back of that? If extensions are wired correctly, when the lower half of the master is removed the extensions should be dead.
If your extensions are wired correctly and it would be acceptable to have your router permanently by the master socket a filtered face plate would be your ideal solution.
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